PDE approach to integration of Heston equations.

e are still investigating the equations
(
Affine equations
),
(
Heston equations
) and aiming to recover the
expression for
were the process
is given by the
equations
and the
are constants,
are increments of independent standard Brownian motions. According to the
general theory of the Backward Kolmogorov's equation, see the section
(
Backward equation section
), we have
the following PDE and initial
condition:
We look for a solution of the
form
We substitute such representation into the
PDE:
To transform the boundary condition we use the inverse Fourier
transform:
The expression of the
form
is Dirac's delta function. Indeed, for any smooth quickly decaying
and Fourier transform
Hence,

We continue with investigation of the
equation
We seek a solution of the
form
We
have
hence
The last equation should be satisfied for every
.
Hence, we separate powers of
:
The above equations are subject to the final
conditions
The expressions for
,
may be obtained with the technique described in the section on the Ricatti
equation (
Ricatti equation
).